Weapon talents exist again upon kills?

Yes.

I'm not sure how it works exactly but after finishing ME1 (MELE) on Soldier, I had Sniper Rifles and Assault Rifles unlocked as bonus powers for the other classes.

I finished ME1 with Vanguard and then Infiltrator. When I fired up ME2, I had both Vanguard and Infiltrator powers and gun classes available to choose from. I think it's just a matter of which class you complete the game with. I might be wrong though.

The achievements for using powers are still there and the bonus powers unlock when those achievements unlock.  The change there is that using a squad mates powers enough times will unlock the achievements and related bonus powers;whereas, the old game required Shepard to use those powers to unlock them.  With my soldier, I unlocked Singularity, Overload, and Medicine as bonus powers for my next run by merely directly Liara and Kaidan to use their powers frequently.  Unlocking these bonus powers with a Soldier Shepard used to be impossible with the old ME1; one had to create an Adept Shepard to get singularity, a sentinel Shepard to get Medicine, and either an engineer or a infiltrator Shepard to get Overload.

Ugh. I hope the bonus weapon trainings simply unlock after completing the game. I don't think I'm likely to play a ME1 Soldier any time soon. With all other bonus skills unlocked after a single playthrough, I'm unlikely to ever play anything but Infiltrator, Sentinel or Vanguard in ME1.

Honestly, I may never even use a bonus skill training on a weapon, but the option to do so is cool.

Good deal. That's what I'd read in other threads. The contradictory info in this thread cast a shadow of doubt.

I'm about halfway through ME3 right now with my Sentinel. For my next Shepard, I'm leaning Infiltrator, but we'll see. I'd usually be inclined to add another Tech skill, even with Assault Rifles available.

ME1 is so much better without the non-proficient accuracy penalties. My Insanity Sentinel got by just fine with no weapon training. I felt even more free to swap situationally, and ignored only the pistol. I leaned heavily on the Assault Rifle and never felt "untrained". I didn't have Overkill and the minor numerical increases, but I had the Sentinel's slew of biotic and tech skills instead. That's a trade I'll gladly make.

Despite having over 100 prior PTs of ME1, I'd never played a Sentinel. The lack of weapon training combined with the penalties nixed it. I could have added a weapon training, of course; but the pistol was so OP that I saw no point. I mostly played Infiltrators and Vanguards.

I just checked, and yes they can.

You didn't used squadmate talents at all. Impressive. I rushed through Therum and Feros just to unlock all bonus powers and then started main playtrough.
Assault Rifles, Sniper Rifles and Shotguns will unlockk like I said earlier, when you complete the game with any class on any difficulty and earn Medal of Honor achievement.

Do you even have to shoot on lower difficulties or you can just beat everything to the death? I never played on anything lower than Insanity.

Sentinel with Warp or Damping as bonus power wrecks ME1 Insanity. Enjoy.